OMWG - Editorial

PROBLEM LINK:

DIFFICULTY:

simple

PREREQUISITES:

basic understanding of graphs, trial or error

PROBLEM:

There is a $n \times m$ dimensional grid. Initially the cells of the grid are uncoloured. You will colour all the cells of the grid one by one colouring a single cell each time. Score obtained for colouring a cell will be equal to number of already coloured neighbouring cells of the cell that you are going to colour. Note that two cells are considered to be neighbours of each other, if they share a side with each other.

QUICK EXPLANATION

The answer will be equal to $n \cdot(m - 1) + m \cdot (n - 1)$.

EXPLANATION:

Let us solve the first subtask before proceeding to the next one. In this subtask, we have $1 \leq n, m \leq 3$. Without loss of generality, we can assume that $n \leq m$, i.e. answer for grid of dimensions $n, m$ where $n \leq m$, will be same as that of a grid of dimensions $m, n$ where $m \leq n$, because both the grids are essentially the same. You can rotate one to obtain the other.

So, we are now left to deal with following cases.

$n = 1, m = 1$ : You can just colour the current cell. Total score will be obtained will be zero.

$n = 1, m = 2$ : Colour any of the cells first. For colouring the second cell, you will get a score of 1.

$n = 1, m = 3$ : There are three cells. There are total 3! ways of choosing the order of the cells to color. Let us find scores for some of them.

Colour the cells in the order, i.e. first cell will be colored first, then second followed by third. Colouring first cell won't fetch you any score, whereas that of second and third cells will fetch you a score of 1 each. Total score obtained will be 2.

Colour the second cell first, then you can choose either to color the first cell or the third cell. Overall you will get a score of 2.

$n = 2, m = 2$ : You will obtain a score of 4.

$n = 2, m = 3$ :

You can color cell (1, 1) first, then cell (2, 1) followed by (1, 2) followed by (2, 2), then (1, 3) followed by (2, 3), i.e. you are colouring cells in the column wise order. Your total score will be 0 + 1 + 1 + 2 + 1 + 2 = 7.

Try some other order you will find that your score will always be 7.

$n = 3, m = 3$ : Total score obtained will be 12.

So, for solving the small subtask, you can just find these values manually and solve the problem. Here is one sample code.

Probably a better way of implementing this way would be encode these values in a two dimensional array and output them. It will save you a lot of if/else conditions which are prone to missing some cases.

Now, you might have realized that somehow the order of coloring the vertices does not matter. Let us find a formal proof of this condition, and later we will understand how to use this to solve the full subtask.

Consider a graph whose vertices correspond to the cells of the grid. There will be an edge between two vertices if their corresponding cells are adjacent.

Now, let us understand the colouring process on this graph instead of the grid. So, colouring a cell is equivalent to colouring a vertex of the graph. As score obtained for coloring a cell is equal to number of coloured neighbours of the cell. This means that score obtained will be number of edges between the current cell and its adjacent coloured cells. So, we wonder whether we can transform our problem in terms of counting edges instead?

Whenever we color an adjacent cell of some already coloured cell, we will mark the edge between those cells. Also marking an edge will give you a score of 1. So, our total score will number of times the edges were marked.

Crucial point to note is that after the end of coloring all cells, all the edges of the graph will be marked. This is simply due to that fact each of the grid cells are coloured.

One interesting fact is that the each edge of the graph will be marked only $once$. This is due to the fact that an edge connects two cells. We mark the cell when both the cells are coloured. We are not allowed to colour any cell more than once, which guarantees that we will mark each cell only once.

So, in whatever order you colour the cells, the number of edges marked are going to remain the same, so your score is also going to be same.

Now we can use this fact to find a solution of the problem. We can just implement this process by iterating over the cells of the matrix in any way which want and maintain which of the cells are coloured and count the corresponding score. Pseudo code follows.

read n, m
colored[n][m]; // Initially none of the cells are coloured.
// We are using row major order. You can however use any order of visiting the cells you want.
score = 0;
for i = 1 to n:
for j = 1 to m:
for all neighbours of cell (i, j):
if (colored[i][j]):
score += 1
print score

We can even find a closed form formula for the score. Let us find total number of edges in a $n \times m$ dimensional grid. We know that sum of degrees of all the vertices of the graph is twice the number of edges. It is easier to find sum of degrees of vertices in our case. All the vertices in the inner boundary of the grid have a degree of 4. The corner vertices on the boundary will have a degree of 2, remaining vertices on the boundary will have degree of 3.

As n increases, difference between two successive matrices (increasing m) increases in an Arithmetic Progression. Similarly, when n is fixed and m increases, the maximum score is also an AP. The first terms and common difference can be figured out by trying out a few cases. Here's the pseudocode of the calculation part.